Lecture 4
More on ReCasting Problems in Terms of Random Variables
Example
1 (13.44 on p.427)
The problem statement is as follows: In a study of the effectiveness of certain
exercises in weight reduction, a group of 16 persons engaged in these exercises for one month and showed the
following results:
Table 1.
Measured weights (lbs.) before and after exercise program
Person Number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Before
211
180
171
214
182
194
160
182
172
155
185
167
203
181
245
146
After
198
173
172
209
179
192
161
182
166
154
181
164
201
175
233
142
The numbers in the table did not arise spontaneously. They were the result of actions. Since there are 32
numbers, there were 32 actions. There are 32 numbers because for each of 16 persons, two actions were
taken.
Q: What are the two actions that were taken for each of the 16 persons?
A:
X
=
measuring a person’s weight before the program
;
Y =
measuring that person’s weight after.
Hence, it is natural to let
X
k
(
Y
k
) denote the act of measuring the
k
th
person’s wgt. before (after) the program.
NOTATION:
There are two ways to denote the 32 random variables in a concise manner:
(i) as two collections (or sets):
16
1
}
{
=
k
k
X
&
16
1
}
{
=
k
k
Y
.
(ii) as two vectors (or, arrays):
X =
]
,
,
,
[
16
2
1
X
X
X
&
Y =
]
,
,
,
[
16
2
1
Y
Y
Y
.
The advantage of the form (ii) is that we can then use matrix operations. For example, a scatter plot of
the
x / y
data is shown below.
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140
160
180
200
220
240
260
140
150
160
170
180
190
200
210
220
230
240
x = weight before program
y = weight after program
Figure 1.
Scatter plot of the measured weights of the 16 persons before versus after the program.
The scatter plot suggests that for any arbitrarily chosen person who might enroll in the program, one might be
able to predict what the act of measuring his/her weight after the program (
Y
) will be in relation to the
measurement of his/her weight before the program (
X
). Let
Y
t
denote the act of predicting what the person’s
weight will be after the program. Since the scatter plot has a relatively
linear
appearance, it is reasonable to use
a
linear prediction model
:
b
X
m
Y
+
=
.
(1)
The model (1) is the
slopeintercept
form for a straight line with slope
m
and intercept
b
. Avery, VERY simple
way to obtain numerical values for these two parameters is to take a ruler and draw a straight line through the
data. This procedure resulted in Figure 2.
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 Spring '09
 DontKnow
 Standard Deviation, Randomness, 16 K, 4 lb, 184.25 16 k

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